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with the same value of the parameter to be estimated. This interval
provides us with an estimate of the precision of the corresponding point
estimate.
We resample with replacement repeatedly from the original sample,
1000 times or so, computing the sample statistic for each bootstrap
sample.
For example, here are the heights of a group of 22 adolescents,
measured in centimeters and ordered from shortest to tallest.
137.0 138.5 140.0 141.0 142.0 143.5 145.0 147.0 148.5 150.0 153.0 154.0
155.0 156.5 157.0 158.0 158.5 159.0 160.5 161.0 162.0 167.5
The median height lies somewhere between 153 and 154 cm. If we
want to extend this result to the population, we need an estimate of
the precision of this average.
Our first bootstrap sample, arranged in increasing order of magnitude
for ease in reading, might look like this:
138.5 138.5 140.0 141.0 141.0 143.5 145.0 147.0 148.5 150.0 153.0 154.0
155.0 156.5 157.0 158.5 159.0 159.0 159.0 160.5 161.0 162.0
Several of the values have been repeated; this is not surprising because we
are sampling with replacement, treating the original sample as a stand-in
for the much larger population from which the original sample was drawn.
The minimum of this bootstrap sample is 138.5, higher than that of the
original sample; the maximum at 162.0 is less than the original, while the
median remains unchanged at 153.5.
137.0 138.5 138.5 141.0 141.0 142.0 143.5 145.0 145.0 147.0 148.5 148.5
150.0 150.0 153.0 155.0 158.0 158.5 160.5 160.5 161.0 167.5
In this second bootstrap sample, again we find repeated values; this time
the minimum, maximum, and median are 137.0, 167.5, and 148.5,
respectively.
The medians of 50 bootstrapped samples drawn from our sample ranged
between 142.25 and 158.25, with a median of 152.75 (see Figure 4.1).
These numbers provide an insight into what might have been had we
sampled repeatedly from the original population.
FIGURE 4.1
Scatterplot of 50 Bootstrap Medians Derived from a Sample
of Heights .
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